It can be proved that there are at least T prime numbers less than N, where the recursive function T is: T = N - N*sum(A005867(i)/A002110(i), i = 0..T(sqrt(N))) This can show for example that at least .16*N numbers are prime less than N for 29^2 > N > 23^2. - Ben Paul Thurston, Aug 23 2010

These numbers are divisible by their digital roots, which makes the sequence a subsequence of A064807. - Ivan N. Ianakiev, Oct 08 2013

The above comment from Parthasarathy Nambi follows from the observation that digit summing produces a congruent number mod 9, so the digital root of any multiple of 3 is a multiple of 3. p(n)# is divisible by 3 for n >= 2. - Christian Schulz, Oct 30 2013

The peaks (i.e., local maximums) in a graph of the number of repetitions (i.e., the tally of values) vs. value, as generated by taking the differences of all distinct pairs of odd prime numbers within a contiguous range occur at regular periodic intervals given by the primorial numbers 6 and greater. Larger primorials yield larger (relative) peaks, however the range must be >50% larger than the primorial to be easily observed. Secondary peaks occur at intervals of those "near-primorials" divisible by 6 (e.g., 42). See A259629. Also, periodicity at intervals of 6 and 30 can be observed in the local peaks of all possible sums of two, three or more distinct odd primes within modest contiguous ranges starting from p(2) = 3. - Richard R. Forberg, Jul 01 2015